I shall consider a rhombus tiling model (equivalently, a dimer model on a hexagonal graph) with a free boundary. The correlation of a small triangular hole in this model will be determined. As I shall explain, this kind of problem features phenomena which are parallel to phenomena in...

The fundamental group is a more or less complete invariant of a 3-dimensional manifold. We will discuss how the purely algebraic property of this group being left-orderable is related to two other aspects of 3-dimensional topology, one geometric and the other essentially analytic....

The reliability of computational models of physical processes has received much attention and involves issues such as validation of the mathematical models being used, the error in any data that the models need and the accuracy of the numerical schemes being used, verification. These issues are...

Leibniz is still alive - and so are his mathematical ideas in the third millennium
2016-09-21

In the current year 2016, we are celebrating the third centennial of Leibniz's death. This is a good occasion to present some highlights in the immense mathematical work of the great universal genius under a modern point of view, and to add some surprising interpretations that might have elapsed the...

In this talk I want to present my point of view on what (higher) representation theory is and why it is useful. I would like to show, using a baby example, how classical representation theory transforms into higher representation theory, in particular, explaining what the latter one is. At the end,...

In this lecture I will recall some classical and not so classical normal form theorems in differential geometry and I will explain a new, far reaching, metric approach to these results using the language of Lie groupoids....

We look at whether monetary decisions constitute a significant macro-finance risk for interest rate options and related implied volatilities. We devise an option-pricing model based on the dynamics of the Federal Reserve’s target rate via a regime-shift approach modeled as discrete Markov...

In 1963, Erdos and Rényi proved the paradoxical result that there is a graph R with the property that, if a countable graph is chosen by selecting edges independently with probability 1/2, the resulting graph is almost surely isomorphic to R. Their proof was a beautiful example of a...

Introduced by R. Schwartz more than 20 years ago, the pentagram map acts on plane n-gons, considered up to projective equivalence, by drawing the diagonals that connect second-nearest vertices and taking the new n-gon formed by their intersections. The pentagram map is a discrete completely...

I will outline the main ideas of the new approach to foundations of practical mathematics which we call univalent foundations. Mathematical objects and their equivalences form sets, groupoids or higher groupoids. According to Grothendieck's idea higher groupoids are the same as homotopy types....

We list and illustrate by examples various sources of uncertainty associated with optimization problems. We then explain the difficulties arising when solving such uncertainty affected problems due to lack of full information on the nature of the uncertainty on one hand, and the likelihood of facing...

The Vehicle Routing Problem holds a central place in distribution management. In the classical version of the problem the aim is to distribute goods to a set of geographically dispersed customers under a cost minimization objective and various side constraints. In recent years a number of...

The classical multivariate Hermite interpolation problem asks for the determination of the dimension of the vector space of polynomials of a given degree d in a given number r of variables having n assigned, sufficiently general, zeroes with given multiplicities. This is trivial for n=1, but quite...

From Numerics to Computational Science and Engineering
2011-12-07

Speaker: Rolf Jeltsch (ETH Zurich, Switzerland)

The birth of the discipline numerics is due to the invention of modern computers in the forties of the last century. The main task was to learn to live with the deficiencies of computers, which are the limited memory space, that only a finite number of arithmetical operations can be carried out and...

For a long time there have been two kinds of mathematical computation: symbolic and numerical. Symbolic computing manipulates algebraic expressions exactly, but it is unworkable for many applications since the space and time requirements tend to grow combinatorially. Numerical computing avoids the...

We consider the inverse problem of discovering the location of point sources from very sparse point measurements in a bounded domain that contains impenetrable obstacles. The sources spread according to a class of linear equations, including the Laplace, heat, and Helmholtz equations, and limited...

A fundamental difficulty in understanding and predicting large-scale fluid movements in porous media is that these movements depend upon phenomena occuring on small scales in space and/or time. The differences in scale can be staggering. Aquifers and reservoirs extend for thousands of meters, while...

Compared to the case of functions of one variable, or to the case of symmetric functions, there are not many algebraic tools to manipulate polynomials in several variables. The symmetric group can greatly help in that matter. Newton found how to transform a discrete set of data into an algebraic...

In the new movie version of "Flatland", the central geometric figure in that film is a cube that rotates about its center in three-space producing a number of central slices in the plane including a square, a rectangle, a rhombus, and a hexagon. What are the analogous central hyperplane...

Alternating sign matrices have been introduced by combinatorists in the 80's as an extension of the notion of permutation matrices. A simple formula for the number of such matrices of size n has been conjectured. The resolution of this conjecture has been a very active subject in the 80's and 90's....

One of the central tenets of signal processing is the Shannon/Nyquist sampling theory: the number of samples needed to
reconstruct a signal without error is dictated by its bandwidth-the length of the shortest interval which contains the support of the spectrum of the signal under study. Very...

The purpose of this talk is to give an overview of some recent work concerning the structural and geometric properties of the
evolution of the forward rate curve in an arbitrage free bond market. The main problems to be discussed are as follows.
1. When is a given forward rate model consistent...

The uses of mathematical images in the seventeenth century, from Galileo to Newton
2008-03-13

From long ago, historians have shown the importance of images (emblems books, jesuit propaganda, etc.) during the seventeenth century. Historians of science also emphasized the role played by Optics and its representations. But there has been quite an absence about the role played by images as well...

Flow control is one of the most challenging and relevant topics connecting the theory of
Partial Differential Equations (PDE) and Control Theory. On one hand the number of
possible applications is huge including optimal shape design in aeronautics. On the other
hand, from a purely mathematical...

What's an infinite dimensional manifold and how can it be useful in anatomy?
2007-09-19

There has been a huge explosion in medical
imaging, in spatial and temporal resolution, in imaging many processes as well as organs.
New mathematical tools have entered the game, which are not so elementary. It's important to have some basic understanding of them, to know how what to expect of...

Singular perturbation systems related to segregation
2007-07-25

Speaker: Luis Caffarelli
(University of Texas at Austin, USA)

I will discuss a singular perturbation system related to segregation and the regularity properties of the limiting solution and its free boundaries.
Area(s): ...

What geometry in the early history of architecture?
2006-10-17

Speaker: Kim Williams

It is commonly said that Euclidean geometry played a large part in the development of architecture, but is this really true? An examination of the two earliest and best known treatises of architecture, On architecture by Vitruvius and the Sketchbooks of Villard de Honnecourt reveal no trace of...

Variational Analysis and Generalized Differentiation: New Trends and Developments
2006-07-05

Speaker: Boris Mordukhovich
(Dep. Mathematics, Wayne State Univ.)

Nonsmooth functions, sets with nonsmooth boundaries, and set-valued mappings naturally and frequently appear in various aspects of analysis. Constrained optimization, calculus of variations and its modern form of optimal control, stochastic and statistical problems, mathematical economics, etc., are...

Homomorphisms of the alternating group A_5 into semisimple groups
2006-06-27

Speaker: George Lusztig (Dept. Math., MIT)

Usually in representation theory one studies homomorphisms of a complicated group G into a simpler group: the general linear group of a vector space. But it is also interesting to replace the general linear group by other algebraic groups such as
symplectic, orthogonal or even exceptional groups....

Mathematical models for cell motion
2006-05-30

Speaker: Benoit Perthame
(??cole Normale Supérieure, Paris)

Several transport-diffusion systems arise as simple models in chemotaxis (motion of bacterias or amebia interacting through a chemical signal) and in angiogenesis (development of capillary blood vessels
from an exhogeneous chemoattractive signal by solid tumors). These systems describe the...

Neuronal Calcium Signaling
2006-04-04

Speaker: Steve Cox (Computational and Applied Math., Rice University)

Calcium, the most important of the second messengers, regulates the synaptic plasticity that is underlies our ability to learn. Calcium enters cells through single-protein channels in the cells' outer membrane. We exploit the ability to dynamically monitor
cytosolic calcium, throughout `intact'...

Lorenz Strange Atractors
2006-02-14

Speaker: Marcelo Viana (IMPA, Brasil)

The Lorenz strange attractor, introduced in the sixties in the context of thermal convection and weather prediction, became a
paradigm chaotic behavior, and a crucial model to try and describe this type of dynamics.
In the nineties there were two fundamental developments:
On the one hand,...

The Poincare Conjecture and Algorithmic Problems in Algebra
2005-09-12

Speaker: Alexey Sossinsky (Indep. Univ. Moscow)

It has been announced that the Poincare
Conjecture (claiming that any simply connected closed compact three-dimensional manifold is homeomorphic to the sphere), one of the most famous problems in mathematics,
has been solved by Grisha Perelman of St.Petersburg.
Although no complete verifiable...

One of the beauties of mathematics is that it can uncover connections between seemingly disparate applications. One of the most fertile grounds for unearthing connections is computational algorithms where one often discovers that an algorithm developed for one application is equally useful in...